Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions

2011 ◽  
Vol 74 (15) ◽  
pp. 5171-5188 ◽  
Author(s):  
N. Mavinga ◽  
M.N. Nkashama
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Parabolic Equations ◽  
Bounded Solutions ◽  
Nonlinear Parabolic

scholarly journals Stability of a Hot Smoluchowski Fluid

2001 ◽  
Vol 08 (01) ◽  
pp. 19-27 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Numerical Range ◽  
Nonlinear Parabolic Equations ◽  
Stationary Solutions ◽  
Small Perturbations ◽  
Material Density ◽  
Nonlinear Parabolic ◽  
The Stability ◽  
Special Case

We study coupled nonlinear parabolic equations for a fluid described by a material density ρ and a temperature Θ, both functions of space and time. In one dimension, we find some stationary solutions corresponding to fixing the temperature on the boundary, with no-escape boundary conditions for the material. For the special case, where the temperature on the boundary is the same at both ends, the linearized equations for small perturbations about a stationary solution at uniform temperature and density are derived; they are subject to boundary conditions, Dirichlet for Θ and no-flow conditions for the material. The spectrum of the generator L of time evolution, regarded as an operator on L2[0,1], is shown to be real, discrete and non-positive, even though L is not self-adjoint. This result is necessary for the stability of the stationary state, but might not be sufficient. The problem lies in the fact that L is not a sectorial operator, since its numerical range is ℂ.

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